Solve 72y^2-176y+64

$$72 y ^ { 2 } - 176 y + 64$$

Factor

$$8\left(y-2\right)\left(9y-4\right)$$

Solution Steps

Factor out 8.

$$8\left(9y^{2}-22y+8\right)$$

Consider 9y^{2}-22y+8. Factor the expression by grouping. First, the expression needs to be rewritten as 9y^{2}+ay+by+8. To find a and b, set up a system to be solved.

$$a+b=-22 ab=9\times 8=72$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 72.

$$-1,-72 -2,-36 -3,-24 -4,-18 -6,-12 -8,-9$$

Calculate the sum for each pair.

$$-1-72=-73 -2-36=-38 -3-24=-27 -4-18=-22 -6-12=-18 -8-9=-17$$

The solution is the pair that gives sum -22.

$$a=-18 b=-4$$

Rewrite 9y^{2}-22y+8 as \left(9y^{2}-18y\right)+\left(-4y+8\right).

$$\left(9y^{2}-18y\right)+\left(-4y+8\right)$$

Factor out 9y in the first and -4 in the second group.

$$9y\left(y-2\right)-4\left(y-2\right)$$

Factor out common term y-2 by using distributive property.

$$\left(y-2\right)\left(9y-4\right)$$

Rewrite the complete factored expression.

$$8\left(y-2\right)\left(9y-4\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$72y^{2}-176y+64=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$y=\frac{-\left(-176\right)±\sqrt{\left(-176\right)^{2}-4\times 72\times 64}}{2\times 72}$$

Square -176.

$$y=\frac{-\left(-176\right)±\sqrt{30976-4\times 72\times 64}}{2\times 72}$$

Multiply -4 times 72.

$$y=\frac{-\left(-176\right)±\sqrt{30976-288\times 64}}{2\times 72}$$

Multiply -288 times 64.

$$y=\frac{-\left(-176\right)±\sqrt{30976-18432}}{2\times 72}$$

Add 30976 to -18432.

$$y=\frac{-\left(-176\right)±\sqrt{12544}}{2\times 72}$$

Take the square root of 12544.

$$y=\frac{-\left(-176\right)±112}{2\times 72}$$

The opposite of -176 is 176.

$$y=\frac{176±112}{2\times 72}$$

Multiply 2 times 72.

$$y=\frac{176±112}{144}$$

Now solve the equation y=\frac{176±112}{144} when ± is plus. Add 176 to 112.

$$y=\frac{288}{144}$$

Divide 288 by 144.

$$y=2$$

Now solve the equation y=\frac{176±112}{144} when ± is minus. Subtract 112 from 176.

$$y=\frac{64}{144}$$

Reduce the fraction \frac{64}{144} to lowest terms by extracting and canceling out 16.

$$y=\frac{4}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2 for x_{1} and \frac{4}{9} for x_{2}.

$$72y^{2}-176y+64=72\left(y-2\right)\left(y-\frac{4}{9}\right)$$

Subtract \frac{4}{9} from y by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$72y^{2}-176y+64=72\left(y-2\right)\times \frac{9y-4}{9}$$

Cancel out 9, the greatest common factor in 72 and 9.

$$72y^{2}-176y+64=8\left(y-2\right)\left(9y-4\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 72

$$x ^ 2 -\frac{22}{9}x +\frac{8}{9} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = \frac{22}{9} rs = \frac{8}{9}$$

Two numbers r and s sum up to \frac{22}{9} exactly when the average of the two numbers is \frac{1}{2}*\frac{22}{9} = \frac{11}{9}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{11}{9} - u s = \frac{11}{9} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{8}{9}

$$(\frac{11}{9} - u) (\frac{11}{9} + u) = \frac{8}{9}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{121}{81} - u^2 = \frac{8}{9}$$

Simplify the expression by subtracting \frac{121}{81} on both sides

$$-u^2 = \frac{8}{9}-\frac{121}{81} = -\frac{49}{81}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{49}{81} u = \pm\sqrt{\frac{49}{81}} = \pm \frac{7}{9}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{11}{9} - \frac{7}{9} = 0.444 s = \frac{11}{9} + \frac{7}{9} = 2$$

Evaluate

$$8\left(y-2\right)\left(9y-4\right)$$